报告摘要: It was a great surprise Hans Lewy in 1957 showed that the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain is not locally solvable. Hormander then proved in 1960 that almost all linear partial differential equations are not locally

solvable. Nirenberg and Treves formulated their famous conjecture in 1970: that condition (PSI) is necessary and sufficient for the localsolvability of differential equations of principal type. Principal type means simple characteristics, and condition (PSI) only involves the sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006.

In this talk, we shall present some necessary condition for the solvability of differential operators that are not of principal type, instead the principal symbol vanishes of at least second order at the characteristics. Then the solvability may depend on the lower order terms, and one can define a condition corresponding to (PSI) on the subprincipal symbol. We show that this condition is necessary forsolvability in some cases. The condition is not always necessary, for example effectively hyperbolic operators are always solvable with anylower order terms.